Most sites will have a laundry area - maybe you could pick a site based on that being a criteria.
I would be concerned about drying times if the weather is bad.
TBH, I wouldn't consider it much of a holiday for me if I was going out of my way to wash and dry nappies. If it wasn't as simple as using a machine in my accomodation I would take disposables. Its different if you have use of a private machine but you may have to wait for a slot, ensure you are about to empty the machine etc
Another thing I would worry about is the smell if it was nice weather, tents get VERY warm in hot sunshine (although that'd be a plus for drying them
for a holiday though.